Attracting cycles for the relaxed Newton’s method

We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n ≥ 2 , we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n . We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p ( z ) = z m − c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one.